Flat Standing Sphere Blow-up Solutions for the Nonlinear Heat Equation

Abstract

In this paper, we prove the existence of a singular standing sphere blow-up solution for the nonlinear heat equation with radial symmetry. This solution develops a finite-time singularity on a fixed-radius sphere and exhibits a flat blow-up profile. Our construction refines the method developed by Merle and Zaag \cite{MZJEMS24} which reduces the infinite-dimensional dynamics to a finite-dimensional problem in radial case. The solution satisfies explicit asymptotics near the singular ring and remains regular elsewhere.